Earthquake Analysis of Arch Dams: Factors to Be Considered

The earthquake-induced stresses in each of the four dams computed under two conditions are first compared, considering the following: (1) dam–foundation-rock interaction; and (2) foundation-rock flexibility only (Figs. 1–4). For these two conditions, the largest arch stress on the upstream or downstream face of the dam is 3.3 MPa (476 psi) versus 5.8 MPa (844 psi) for Deadwood Dam; 5.0 MPa (730 psi) versus 9.7 MPa (1410 psi) for Monticello Dam; 4.6 MPa (665 psi) versus 9.2 MPa (1336 psi) for Morrow Point Dam; and 5.2 MPa (758 psi) versus 15.2 MPa (2204 psi) for Hoover Dam. These results demonstrate that if only foundation-rock flexibility is considered, the stresses are overestimated by a factor of approximately 2 for the first three dams and by a factor of approximately 3 for Hoover Dam.

Dam–foundation rock interaction lowers the fundamental resonant frequency of the dam and increases the overall damping in the system. The lowering of the frequency is almost entirely attributable to foundation flexibility with negligible influence of foundation mass and damping (Tan and Chopra 1995a). However, the computed response is larger if only foundation flexibility is considered, because the increase in overall damping caused by foundation damping—material and radiation—is ignored. This energy loss cannot be modeled by increasing the damping value for the dam in a dam-foundation model that considers foundation flexibility only, because there is no rational basis (short of full dam-foundation interaction analysis) to determine the quantitative increase in damping.

Because such overestimation of stresses may lead to overly conservative designs of new dams and to the erroneous conclusion that an existing dam is unsafe and thus requires upgrading, it is imperative that dam–foundation-rock interaction effects be included in earthquake analysis of concrete dams.

Next, the earthquake-induced stresses in two of the four dams are compared. The stresses are computed under two conditions: with water compressibility considered and with water compressibility neglected. In the latter case, reservoir boundary absorption effects are implicitly neglected. By neglecting water compressibility and reservoir boundary absorption, the stresses may be significantly underestimated, as in the case of Monticello Dam (Fig. 5), or considerably overestimated, as in the case of Morrow Point Dam (Fig. 6); note that these discrepancies vary with the location on the dam surface. Thus, water compressibility and reservoir boundary absorption should be included in the analysis of arch dams.

It is not possible to identify the system parameters for which dam response will be overestimated (or underestimated) by neglecting water compressibility. This is because water compressibility modifies the resonant frequency, increases the overall damping (because of the radiation of energy upstream or through the reservoir bottom), and significantly influences the shape of the frequency response curve (Tan and Chopra 1995a). Thus, the influence of water compressibility on the earthquake response of a dam would depend on the frequency characteristics of the ground motion as well as the vibration periods of the dam, increasing the response in some cases but reducing it for others, as observed previously.

The quasi-static component is a significant but not dominant part of the displacement response of Pacoima Dam to the January 13, 2001, earthquake records. Fig. 8, which identifies the quasi-static component in the history of displacements at the center of the crest, shows that the quasi-static component is not a major part of the displacement in the radial direction (the direction of largest response), whereas it is dominant in the displacements in the tangential and vertical directions. At the crest center, the ratio of the peak values of the quasi-static component and the total displacement is 78% for the tangential response and 90% for the vertical response, but only 45% for the radial response.

Therefore, the spatial variations in ground motions are expected to significantly influence, but not dominate, the stresses in the dam, as shown in Fig. 9, which presents the peak values of the tensile stresses in the cantilever direction on the downstream face of the dam; similar figures for arch and cantilever stresses on both faces of the dam are available in Chopra and Wang (2008). Presented are stresses caused by four different excitations. The first three are spatially uniform excitations defined by ground motions recorded at the base of the dam (Channels 9–11), the right abutment (Channels 12–14), and the left abutment (Channels 15–17); see Fig. 7. The fourth excitation is defined as the recorded (and interpolated or extrapolated) spatially varying ground motions.

The stresses caused by spatially varying excitation may be smaller or larger than those caused by spatially uniform ground motion, depending on the intensity of the latter. They are larger when compared to the stresses caused by the base motion—the least intense of the three spatially uniform excitations—but are generally smaller than the stresses caused by the abutment motions. Comparing the four parts of Fig. 9 also reveals that spatial variations in ground motion significantly influence, but not dominate, the stresses in Pacoima Dam caused by the January 13, 2001, earthquake.

The quasi-static component is dominant in the displacement response of Pacoima Dam to the January 17, 1994, Northridge earthquake records; the spatial variations in ground motion therefore have profound influence on the computed stresses in the dam. Fig. 10 identifies the quasi-static component in the history of displacements at the center of the dam crest. The ratio of the peak values of the quasi-static component and the total displacement at the crest center is 87, 93, and 97% for the radial, tangential, and vertical directions, respectively. Consequently, the spatial variations in ground motion are expected to profoundly influence the stresses in the dam. Fig. 11, which presents the peak value of the tensile stresses in the arch direction on the downstream face of the dam, confirms this expectation; similar figures for arch and cantilever stresses on both faces of the dam are available in Chopra and Wang (2008). Presented are the stresses caused by four different excitations. The first three are spatially uniform excitations defined by ground motion at the base of the dam (Channels 9–11), the right abutment (Channels 12–14), and the left abutment (Channels 15–17); see Fig. 7. The fourth excitation is defined as the “recorded” (and interpolated or extrapolated) spatially varying ground motion. The distribution pattern of stresses caused by the three spatially uniform excitations is similar, although the magnitude of stresses caused by ground motions recorded at the base of the dam is much smaller than that caused by motions at the left or right abutment; the stresses caused by the two abutment excitations are similar in magnitude. Comparing the stresses caused by spatially varying and spatially uniform excitations shows that spatial variations in ground motion had a profound influence on the magnitude and the distribution of arch stresses (Fig. 11). In particular, note the stress concentration near the top of the left abutment [Fig. 11(d)]. Spatial variations in ground motion cause much larger cantilever stresses on both faces (compared with all three spatially uniform excitations) in portions of the dam adjacent to the dam–foundation rock contact (Chopra and Wang 2008).

Earthquake records from M3.6 to M4.9 earthquakes centered 12–20 km away from three well-instrumented arch dams in Switzerland provided a rare set of data, although for very low intensity ground motion, for evaluating whether it is reasonable to assume the foundation rock as massless, as is common in engineering practice. Finite-element models of the dam-–foundation rock system, assuming foundation rock to be massless, were developed and their properties calibrated against ambient tests and forced vibration tests; this calibration led to damping ratios of 2–3% for the dams (Proulx et al. 2004). The response of the Mauvoisin and Punt-dal-Gall dams caused by ground motion recorded in the free field away from the dam base, and of the Emosson Dam caused by base motion, were computed. Although the models were calibrated against data from ambient or forced vibration tests, the computed motions at the dam crest were much larger than the earthquake records. To achieve reasonable agreement between computed and recorded motions, the damping ratio had to be increased to 15% at Emosson and to 8% at Mauvoisin and Punt-dal-Gall. Because these damping values are unrealistically large, it was concluded that the assumption of massless foundation rock in the preceding analysis that implied neglecting foundation material and radiation damping was unrealistic. Using the calibrated damping ratio of 2–3% for the dam and reasonable damping values for the foundation rock, subsequent analysis included dam foundation–rock interaction effects (considering foundation inertia, material damping, and radiation damping). These analyses, implemented on the EACD-3D-96 computer program, led to computed responses that were closer to the measured responses, but significant discrepancies remained.

Recently, the response of the Mauvoisin Dam to the recorded spatially varying ground motion was computed by the EACD-3D-2008 program and compared with recorded motions. A finer finite-element model was developed, and the damping properties of the dam and foundation rock were selected to calibrate with the overall damping of 2–3% for the dam-water-foundation-rock system measured from forced vibration tests. The computed displacement responses of the dam are reasonably similar to the recorded displacements, but the agreement is far from perfect. The peak values of the computed displacements in the stream direction—the direction of the largest response—are very close to the recorded value (Fig. 12); at a node near the crest center it is 92% of the recorded value, and at a node near the crest right quarter point it is 101% of the recorded value. However, the computed displacement history does not agree as well; although it is very close to that recorded over some time segments, the two differ significantly during other time segments (Fig. 12).

Recognizing that the recorded ground motions provide an incomplete description of the earthquake excitation and that no attempts were made to adjust the published data for parameter values for the mass and stiffness properties of the concrete and rock (Proulx et al. 2004), the agreement between computed and recorded motions is good considering the complexity of the system analyzed.

The response of the Pacoima Dam to the spatially varying ground motions recorded during the 2001 earthquake is determined by the EACD-3D-2008 computer program. The computed displacements compare well with the recorded displacements (Fig. 13); the agreement is much better than that obtained earlier in the case of Mauvoisin Dam. The time variation of the computed displacements is close to the recorded response over its entire duration; however, the peak displacement at Channels 1 and 2 is overestimated. Recognizing that the recorded ground motions provide only an incomplete description of the earthquake excitation and that no attempts were made to adjust the published data (Alves 2004; Alves and Hall 2006) for the mass and stiffness parameters of concrete and rock, the agreement between computed and recorded displacements is satisfactory.

A similar comparison of responses computed by linear analysis against motions recorded during the Northridge earthquake is not appropriate because the ground shaking was intense enough to cause significantly nonlinear behavior in the dam, as manifested by the opening of vertical contraction joints and the cracking of concrete. However, a linear analysis by the EACD-3D-2008 computer program predicted large arch stresses in the thrust block between the dam and the left abutment and in the portion of the dam adjacent to the thrust block [see Fig. 11(d) and additional figures in Chopra and Wang (2008)], suggesting that cracking would occur in these areas, which is what actually happened during the earthquake; see Fig. 14.